Solve each quadratic inequality. Give the solution set in interval notation.
(a) -(x + 1)(x + 2) ≥ 0
(b) -(x + 1)(x + 2) > 0
(c) -(x + 1)(x + 2) ≤ 0
(d) -(x + 1)(x + 2) < 0

Verified step by step guidance

1. First, rewrite the inequality in a clearer form. The expression is \(-(x + 1)(x + 2)\), so identify the critical points by setting the expression inside the parentheses equal to zero: solve \(x + 1 = 0\) and \(x + 2 = 0\) to find \(x = -1\) and \(x = -2\).
2. Next, determine the sign of the expression \(-(x + 1)(x + 2)\) on the intervals defined by the critical points. These intervals are \((-\infty, -2)\), \((-2, -1)\), and \((-1, \infty)\). Choose test points from each interval and substitute them into the expression to check whether the expression is positive, negative, or zero.
3. For each inequality (a) \(\geq 0\), (b) \(> 0\), (c) \(\leq 0\), and (d) \(< 0\), use the sign information from the test points to determine which intervals satisfy the inequality. Remember to include or exclude the critical points depending on whether the inequality is strict or not.
4. Write the solution sets for each inequality in interval notation. For example, if the expression is positive on \((-\infty, -2)\) and \((-1, \infty)\) and zero at \(x = -2\) and \(x = -1\), then for \(\geq 0\) you include the points where the expression equals zero, but for \(> 0\) you exclude them.
5. Finally, verify your solution by considering the graph of the quadratic expression \(-(x + 1)(x + 2)\), which is a parabola opening downward (due to the negative sign). This helps confirm the intervals where the expression is positive or negative.

Key Concepts

Factoring Quadratic Expressions

Factoring involves expressing a quadratic expression as a product of two binomials. In this problem, the quadratic is already factored as -(x + 1)(x + 2), which helps identify the roots and analyze the sign of the expression. Recognizing factors is essential for solving inequalities involving quadratics.

Solving Quadratic Inequalities

Solving quadratic inequalities requires determining where the quadratic expression is positive, negative, or zero. This involves finding critical points (roots), testing intervals between these points, and using the inequality sign to select the correct solution set. The solution is often expressed in interval notation.

Interval Notation and Number Line Analysis

Interval notation concisely represents sets of numbers satisfying inequalities. Using a number line, one tests values in intervals defined by roots to determine where the inequality holds. Understanding how to write and interpret intervals is crucial for expressing the solution set clearly.